Thomson scattering: One rate to rule them all

The enduring tension between local and distant measurements of $H_0$ remains unresolved. It was recently pointed out that cosmic microwave background (CMB) and large-scale structure (LSS) observables are invariant under a uniform rescaling of the gravitational free-fall rates of all species present and the Thomson scattering rate between photons and electrons. We show that a unique variation of the fine-structure constant $\alpha$ and the electron mass $m_{\rm e}$ can leverage this scaling transformation to reconcile the CMB and LSS data with a broad spectrum of Hubble constant values, encompassing those inferred from local measurements. Importantly, this study demonstrates that the constraints on the variation of fundamental constants imposed by the specific recombination history are not as stringent as previously assumed. Our work highlights the critical role of the Thomson scattering rate in the existing Hubble tension and offers a distinct avenue of exploration for particle model builders.